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arXiv:1208.5193v1 [math.QA] 26 Aug 2012

Categorical aspects of compact quantum groups

Alexandru Chirvasitu∗

August 28, 2012

Abstract

We show that either of the two reasonable choices for the category of compact quantum

groups is nice enough to allow for a plethora of universal constructions, all obtained “by abstract

nonsense” via the adjoint functor theorem. This approach both recovers constructions which

have appeared in the literature, such as the quantum Bohr compactification of a locally compact

semigroup, and provides new ones, such as the coproduct of a family of compact quantum

groups, and the compact quantum group freely generated by a locally compact quantum space.

In addition, we characterize epimorphisms and monomorphisms in the category of compact

quantum groups.

Keywords: compact quantum group, CQG algebra, presentable category, SAFT category, adjoint

functor theorem

Contents

1Preliminaries

1.1Compact quantum groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2CQG algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3Locally compact and algebraic quantum spaces and semigroups . . . . . . . . . . . .

1.4SAFT categories and the adjoint functor theorem . . . . . . . . . . . . . . . . . . . .

1.5Finitely presentable categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

4

5

7

8

9

2The category of CQG algebras is finitely presentable10

3

C*QG is SAFT14

4 Applications

4.1Limits in CQG and C*QG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2Quantum groups generated by quantum spaces . . . . . . . . . . . . . . . . . . . . .

4.3Variations on the Bohr compactification theme . . . . . . . . . . . . . . . . . . . . .

4.4Kac quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

16

20

21

23

5Monomorphisms 25

References 26

∗UC Berkeley, chirvasitua@math.berkeley.edu

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Introduction

Compact quantum groups were introduced in essentially their present form in [Wor87] (albeit under

a different name), and the area has been expanding rapidly ever since. The subject can be viewed as

part of Connes’ general program [Con94] to make “classical” notions (spaces, topology, differential

geometry) non-commutative: One recasts compact groups as C∗-algebras via their algebras of

continuous functions, and then removes the commutativity assumption from the definition (see

§1.1 for details). Starting with the relatively simple resulting definition, all manner of compact-

group-related notions and constructions can then be generalized to the non-commutative setting:

Peter-Weyl theory ([Wor87, Wor98]), Tannaka-Krein duality and reconstruction ([Wor88, Wan97]),

Pontryagin duality ([PW90]), actions on operator algebras ([Boc95, Wan98, Wan99]) and other

structures, such as (classical or quantum) metric spaces ([Ban05, QS10]) or graphs ([Bic03]), and

so on. This list is not (and cannot be) exhaustive.

The goal of this paper is to analyze compact quantum groups from a category-theoretic per-

spective, with a view towards universal constructions.

Bits and pieces appear in the literature: In [Wan95], Wang constructs coproducts in the category

C*QG (see §1.1), opposite to that of compact quantum groups (it consists of C∗-algebras, which are

morally algebras of functions on the non-existent quantum groups). More generally, he constructs

other types of colimits (e.g. pushouts) of diagrams with one-to-one connecting morphisms. The

fact that this coproduct can be constructed simply at the level of C∗-algebras (forgetting about

comultiplications) parallels the fact that classically, the underlying space of a categorical product

?Gi of compact groups Gi is, as a set, just the ordinary Cartesian product. The category of

compact groups, however, also admits coproducts, and they are slightly more difficult to construct:

One endows the ordinary, discrete coproduct?Gi(i.e. coproduct in the category of discrete groups,

also known as the free product of the Gi) with the finest topology making the canonical inclusions

Gj →

group. It is natural, then, to ask whether or not coproducts of compact quantum groups exist,

or equivalently, whether the category C*QG opposite to that of compact quantum groups has

products. We will see in §4.1 that this is indeed the case, and moreover, the category is complete

(i.e. it has all small limits).

Another example of universal construction that fits well within the framework of this paper is

the notion of quantum Bohr compactification [So? l05]. One of the main results of that paper is,

essentially, that the forgetful functor from compact quantum groups to locally compact quantum

semigroups has a left adjoint; remembering that we are always passing from (semi)groups to algebras

of functions and hence reversing arrows, this amounts to the existence of a certain right adjoint

([So? l05, 3.1,3.2]). Section 4 recovers this as one among several right-adjoint-type constructions,

such as compact quantum groups “freely generated by a quantum space” (as opposed to quantum

semigroup; see §4.2).

Most compact-quantum-group-related universal constructions in the literature seem to be of a

“left adjoint flavor”: the already-mentioned colimits in C*QG, the quantum automorphism groups

of, say, [Wan98], which are basically initial objects in the category of C*QG objects endowed with a

coaction on a fixed C∗-algebra, etc. By contrast, apart from the Bohr compactification mentioned in

the previous paragraph, universal constructions of the right adjoint flavor (limits in C*QG, or right

adjoints to functors with domain C*QG) appear not to have received much attention. This is all the

more surprising for at least two reasons. First, they seem to be more likely to exist than the other

kind of universal construction; example: a (unital, say) C∗-algebra A endowed with a coassociative

map A → A⊗A into its minimal tensor square (this would be the object dual to a compact quantum

semigroup) always has a compact quantum group (meaning its dual object, as in Definition 1.1.1)

?Gi continuous, and then takes the Bohr compactification of the resulting topological

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mapping into it universally, but does not, in general, have a compact quantum group receiving

a universal arrow from it (Remark 4.3.3). Secondly, the representation-theoretic interpretation of

limits in C*QG is often simpler than that of colimits; see Proposition 4.1.3 (especially part (a))

and surrounding discussion.

The structure of the paper is as follows:

Section 1 recalls the machinery that will be used in the sequel and fixes notations and con-

ventions, introducing the two versions of the category opposite that of compact quantum groups:

CQG, consisting of so-called CQG algebras (these are like the algebra of representative functions

on a compact group; see Definition 1.2.2), and C*QG, whose objects are analogous to algebras of

continuous functions on compact groups (Definition 1.1.1).

In Section 2, Theorem 2.0.5 shows that the category CQG is finitely presentable (§1.5). This

technical property will later allow us to reduce the existence of right adjoints for functors defined

on CQG to checking that these functors are cocontinuous, i.e. preserve colimits. This is typically

an easy task, as the routine nature of most proofs in Section 4 shows.

Section 3 proves a property slightly weaker than finite presentability for the category C*QG

(Theorem 3.0.13). The nice features from the previous section are preserved however, and the

same types of results (existence of right adjoints to various functors defined on C*QG) follow.

In Section 4 we list some of the consequences of the previous two sections. These include the

automatic existence of limits in CQG and C*QG (§4.1), compact quantum groups freely generated

by quantum spaces and semigroups (§§ 4.2 and 4.3 respectively), and universal Kac type compact

quantum groups associated to any given compact quantum group (§4.4).

Finally, in Section 5 we characterize monomorphisms in the categories CQG and C*QG. It

turns out that in the former they have to be one-to-one, whereas in the latter being mono is

slightly weaker than injectivity (Proposition 5.0.3). The results are analogous to the fact ([Rei70,

Proposition 9]) that epimorphisms of compact groups are surjective.

Acknowledgements

This work is part of my PhD dissertation. I would like to thank my advisor Vera Serganova for all

the support, and Piotr So? ltan for helpful discussions on the contents of [So? l05].

1 Preliminaries

All algebraic entities in this paper (algebras, coalgebras, bialgebras, etc.)

algebra is, as usual, a complex algebra endowed with a conjugate linear, involutive, algebra anti-

automorphism ‘∗’. Unless we are dealing with non-unital C∗-algebras as in §1.3 below, in which case

the reader will be warned, algebras are assumed to be unital (and coalgebras are always counital).

Our main references for the necessary basics on coalgebra, bialgebra and Hopf algebra theory

are [Abe80, Mon93, Swe69]. The notation pertaining to coalgebras is standard: ∆ for antipodes

and ε for the counit, perhaps adorned with the name of the coalgebra if we want to be more precise

(example: ∆C, εC). The same applies to antipodes for Hopf algebras, which are usually denoted

by S. We use Sweedler notation both for comultiplication, as in ∆(c) = c1⊗ c2, and for comodule

structures: If ρ : M → M⊗C is a right C-comodule structure, it will be written as ρ(m) = m0⊗m1.

All comodules are right, and the category of right comodules over a coalgebra C is denoted by MC.

For any comodule V over any coalgebra H (the notation suggests that it will become a Hopf

algebra soon), there is a largest subcoalgebra H(V ) over which V is a comodule. If the comodule

structure map is ρ : V → V ⊗ H and (ei)i∈Iis a basis for V ,then H(V ) is simply the span of the

are complex.A ∗-

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elements uijdefined by

ρ(ej) =

?

i

ei⊗ uij.

We refer to uijas the coefficients of the basis (ei), to (uij) as the coefficient matrix of the basis,

and to H(V ) as the coefficient coalgebra of V . The coalgebra structure is particularly simple on

coefficients:

∆(uij) =uik⊗ ukj,

?

k

ε(uij) = δij.

Henceforth, the standing assumption whenever we mention coefficients and coefficient coalgebras

is that the comodule in question is finite-dimensional (that is, I is finite). When V is simple, the

coefficients uijwith respect to some basis are linearly independent, and the coefficient coalgebra is

a matrix coalgebra, in the sense that its dual is a matrix algebra.

1.0.1 Remark Note that maps V → V ⊗C are the same as elements of V ⊗V∗⊗C = End(V )⊗C.

If u = (uij) is the coefficient matrix of a basis ei, i = 1,n for V and End(V ) is identified with Mn

via the same basis ei, then the element of End(V ) ⊗ C∼= Mn(C) corresponding to the coaction

is exactly the coefficient matrix u. We will often blur the distinction between these two points of

view, and might refer to u itself as the comodule structure.

?

If in the above discussion H is a Hopf algebra, more can be said: The matrix S(uij)i,jis inverse

to (uij)i,j. Moreover, giving the dual V∗the usual right H-comodule structure

?f0,v?f1= ?f,v0?S(v1),v ∈ V, f ∈ V∗,

the coefficient matrix of the basis dual to (ei) is precisely (S(uji))i,j(note the flipped indices).

A word on tensor products: In this paper, the symbol ‘⊗’ means at least three things. When

appearing between purely algebraic objects, such as algebras or just vector spaces, it is the usual,

algebraic tensor product. Between C∗-algebras it always means the minimal, or injective tensor

product ([Tak02, IV.4]). Finally, on rare occasions, we use the so-called spatial tensor product

(referred to as W∗-tensor product in [Tak02, IV.5]) between von Neumann (or W∗) algebras. It

will always be made clear what the nature of the tensored objects is, so that no confusion is likely

to arise.

1.1 Compact quantum groups

This is by now a very rich and well-referenced theory, so we will be very brief, and will refer the

reader to one of the many excellent sources (e.g. the papers and book cited below and the references

therein) for details on the topic.

No list of references would be complete without mentioning the seminal papers [Wor87, Wor88],

where Woronowicz laid the foundation of the subject, introducing the main characters under the

name “compact matrix pseudogroups”, while an exposition of the main features of the theory is

given by the same author in [Wor98]. Other good references are the survey paper [KT99], and

[KS97, 11.4].

As mentioned in the introduction, the main idea is that since one can study compact groups by

means of the algebras of continuous functions on them, which are commutative, unital C∗-algebras

with some additional structure, dropping the commutativity assumption but retaining the extra

structure should still lead to interesting objects, which are trying to be “continuous functions on

a quantum group”. The additional structure just alluded to is captured in the following definition

([KT99, 3.1.1]):

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1.1.1 Definition A compact quantum group is a pair (A,∆), where A is a unital C∗-algebra, and

∆ : A → A ⊗ A is a morphism of unital C∗-algebras satisfying the conditions

1. (Coassociativity) (∆ ⊗ id) ◦ ∆ = (id⊗∆) ◦ ∆;

2. (Antipode) The subspaces ∆(A)(1 ⊗ A) and ∆(A)(A ⊗ 1) are dense in A ⊗ A.

C*QG is the category whose objects are compact quantum groups, and whose morphisms f :

A → B are unital C∗-algebra maps preserving the comultiplication in the sense that (f ⊗f)◦∆A=

∆B◦ f.

?

The second condition needs some explanation. The space ∆(A)(1 ⊗ A) is defined as the linear

span of products of the form ∆(a)(1 ⊗ b) ∈ A ⊗ A, and similarly for ∆(A)(A ⊗ 1). The condition

is named ‘antipode’ because it has to do with the demand that A, regarded as a kind of bialgebra,

have something like an antipode. For comparison, consider the case of an ordinary, purely algebraic

bialgebra B. The condition that it have an antipode, i.e. that it be a Hopf algebra, is equivalent

to the requirement that the map

B ⊗ B → B ⊗ B,x ⊗ y ?→ x1⊗ x2y

be a bijection.

Since compact quantum groups as defined above are morally functions of algebras, representa-

tions of the group must be comodules of some sort over the corresponding algebras (endowed with

their comultiplication). Some care must be taken, as Definition 1.1.1 makes no mention of a counit,

and so the usual definition of comodule has to be modified slightly. The solution is (see [KT99,

discussion before Proposition 3.2.1]):

1.1.2 Definition Let (A,∆) be a compact quantum group. A finite-dimensional comodule over A

is a finite-dimensional complex vector space V together with a coassociative coaction ρ : V → V ⊗A

such that the corresponding element of End(V ) ⊗ A is invertible.

?

We will often drop the adjective ‘finite-dimensional’. Remark 1.0.1 applies, and we will often

refer to the coefficient comatrix of some basis as being the comodule structure. [KS97, 11.4.3,

Lemma 45] says that any comodule is unitarizable, in the sense that there is an inner product on

V making the coefficient matrix u ∈ Mn(A) of an orthonormal basis unitary (cf. Definition 1.2.1).

1.2CQG algebras

These are the algebraic counterparts of compact quantum groups. More precisely, if a compact

quantum group as in Definition 1.1.1 plays the role of the algebra of continuous functions on a

“quantum group”, then the associated CQG algebra wants to be the algebra of representative func-

tions of the quantum group, i.e. matrix coefficients of finite-dimensional unitary representations.

The main reference for this subsection is [KS97, 11.1-4].

Recall that a Hopf ∗-algebra H is a Hopf algebra with a ∗-structure making H into a ∗-algebra,

and such that the comultiplication and counit are morphisms of ∗-algebras. This is the kind of

structure that allows one to define what it means for a representations of a quantum group (i.e. a

comodule over the corresponding “function algebra”) to be unitary.

Let V be an n-dimensional comodule over a Hopf ∗-algebra H.

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